Crystals are materials composed of elements arranged in a repetitive order that confers on the overall structure a wavelike configuration. These “waves” tend to interact with wavelike excitations traversing such structures. For example, propagation of a vibration through such a structure, although not straightforward, can be fully and predictably described (as a phonon) given sufficient knowledge of the wave-structure of the material. Photonic crystals comprise two or more periodically repeating dielectric materials with which photons (i.e., electromagnetic waves, which is the more straightforward choice of name for the wave/particle duality in this context) interact according to the laws of refraction that apply to electromagnetic waves according to Maxwell's equations. Photonic crystals have numerous applications as efficient radiation sources, sensors and optical computer chips.
A consequence of the repetitious structure of photonic crystals is that light of certain wavelengths (or “frequencies,” the reciprocal of wavelength) traveling in certain directions and orientations (or “polarizations”) will not propagate through the crystal. That is, across the entire spectrum of wavelengths, certain frequency ranges (or “bands”) cannot pass through the structure. The transmitted spectrum thus has “gaps” in it. In common parlance, the property of the structure that gives rise to a gap in the transmitted spectrum is called a “band gap.” Band gaps that reject a band of frequencies no matter their direction or polarization are called “complete band gaps.” The utility of such band gaps, like the electronic band gaps in semiconductors, lies in their susceptibility to being breached by defects intentionally introduced into the structure. By violating the perfect periodic arrangement of the dielectric material-elements of the crystal, a previously prohibited frequency band is allowed passage into the crystal, where it may be trapped, re-directed, or otherwise altered.
The usefulness of the complete band gaps in periodic structures is, however, limited by the very periodicity on which they depend. Periodicity limits the engineer of photonic crystals to symmetries that do not lend themselves to dielectric materials other than those with dielectric constants that produce very high dielectric contrast. For many applications, furthermore, the band gaps in periodic structures tend to be of limited usefulness because their anisotropy makes devices made with them highly direction-dependent. Periodicity also limits the engineer to a narrow choice of defects, and increases the risk of introducing unintended defects during fabrication. These periodicity-induced limitations apply also to structures sized to control the passage of phonons and electrons.
Materials are needed that relax constraints imposed by periodicity but allow the artisan to engineer complete band gaps that preferably do not depend on direction or polarization.